Rényi showed that there is a \(T_{\beta }\) -preserving and ergodic measure \(\mu _{\beta }\)  such that \(1 - \frac{1}{\beta } \le \frac{d \mu _{\beta }}{d \lambda } \le \frac{1}{1 -1 / \beta }\) . (See [1]} and also see [2]} and [3]}.)
